An extension theorem for weak solutions of the 3d incompressible Euler equations and applications to singular flows

We prove an extension theorem for local solutions of the 3d incompressible Euler equations. More precisely, we show that if a smooth vector field satisfies the Euler equations in a spacetime region Ω × (0, T), one can choose an admissible weak solution on R3 × (0, T) of class Cβ for any β locβ with the same vortex sheet initial data, which coincide with it at each time t outside a turbulent region of width O(t). Second, given any smooth solution v of the Euler equation on T3 × (0, T) and any open set U ⊂ T3, we construct admissible weak solutions which coincide with v outside U and are uniformly close to it everywhere at time 0, yet blow up dramatically on a subset of U × (0, T) of full Hausdorff dimension. These solutions are of class Cβ outside their singular set. © The Author(s), 2025. Published by Cambridge University Press.